If the system of equations
(λ-1) x+(λ-4) y+λ z=5 ; λ x+(λ-1) y+(λ-4) z=7 ; (λ+1) x+(λ+2) y-(λ+2) z=9
has infinitely many solutions, then λ²+λ is equal to

Solution

Maths sir · 4 steps

Add the first and third equations and subtract twice the second: almost everything cancels.

1. Combine the equations. Do E₁ + E₃ - 2E₂. The x terms give 2λ - 2λ = 0, the y terms give (2λ - 2) - (2λ - 2) = 0, and the right sides give 5 + 9 - 14 = 0.

   (6 - 2λ) z = 0

2. Determinant. The same row operation leaves the coefficient determinant unchanged, so Δ = (6 - 2λ)[λ(λ+2) - (λ-1)(λ+1)] = (6 - 2λ)(2λ + 1). Infinitely many solutions need Δ = 0, so λ = 3 or λ = -1/2.
3. Test each value. At λ = -1/2 the combined equation forces z = 0. The second equation then gives x + 3y = -14, but the third gives x + 3y = 18, so there is no solution. At λ = 3 the combined equation becomes 0 = 0, leaving 3x + 2y - z = 7 and 4x + 5y - 5z = 9. These are not proportional, so there are infinitely many solutions.
4. So. λ = 3, and λ² + λ = 9 + 3 = 12, option (d).

The idea to keep. Δ = 0 only says the system is not uniquely solvable. Check each root: it gives infinitely many solutions only if the equations do not contradict each other.

Answer from the NTA final answer key. Solution by PrepWiser.